What is a Linear Equation

Let us consider a small puzzle: Think of a number and add 7 to get 12. What is the number?
We can easily say that the number must be 5.
If we use a letter, i.e., (variable) to stand for the unknown number, we can write this puzzle as follows:
x + 7 = 12 …(1)
Here, if x = 5, 5 + 7 = 12. So, x = 5 is the unknown number which satisfies the statement (1). This is nothing but an equation. An equation is a mathematical statement equating two expressions. The expression on the left side of the equal sign is called LHS (Left Hand Side) and the expression on the right side of the equal sign is called RHS (Right Hand Side). The expressions on either side of equal sign are called members of the equation.

An equation which involves a variable with the highest power 1 is called a linear equation.
Examples: 2x + 3 = 7, x + y = 9, a + b = 2.5 are examples of a linear equation.
But, what about x² + 4 = 13? Is it a linear equation? No because variable x has its power 2.

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Example 1: Write the following mathematical statements as algebraic equations:
(a) 3 more than 5 times of x gives 18.
(b) The sum of p and 7 gives 12.
(c) One third of a number increased by 5 is 8.
Solution:
(a) Five times x = 5x
3 more than five times x = 5x + 3.
It is 18.
∴ The equation is 5x + 3 = 18.
(b) Sum ofp and 7 = p + 7.
It is 12.
∴ The equation is p + 7 = 12
(c) Let the number be x.
One-third of x = 1/3 of x = x/3
One-third of x increased by 5 = x/3 + 5.
It is 8.
∴ The equation is x/3 + 5 = 8

Example 2: Convert the following equations into statements:
(a) x + 2 = 5 (b) 5p + 3 = 28 (c) 2/3 m – 1 = 5
Solution:
(a) The sum of x and 2 gives 5.
(b) 3 more than five times of p gives 28.
(c) 1 less than two-third of m is 5.

EQUATIONS WITH A PAIR OF SCALES

An equation behaves like a pair of balanced scales. Both sides of the equation are balanced in the same manner as the scales of a balance. In order to main-tain balance, we need to remember some rules.
balancing-linear-equation Rules for balancing linear equations
Let us consider 6 + 2 = 8. It is an equality, because both of its sides (LHS and RHS) equal to 8.
Rule 1: If we add the same quantity on both sides of an equality, the equality holds true.
6 + 2 = 8
6 + 2 + 3 = 8 + 3   (adding 3 on both sides)
11 = 11 (LHS = RHS)
Rule 2: If we subtract the same quantity from both sides of an equality, the equality holds true.
6 + 2 = 8
6 + 2 – 3 = 8 – 3   (subtracting 3 from both sides)
5 = 5 (LHS = RHS)
Rule 3: If we multiply both sides of an equality by the same quantity, the equality holds true.
6 + 2 = 8
(6 + 2) × 3 = 8 × 3   (multiplying by 3 on both sides)
6 × 3 + 2 × 3 = 8 × 3
18 + 6 = 24
24 = 24 (LHS = RHS)
Rule 4: If we divide both sides of an equality by the same quantity, the equality holds true.
6 + 2 = 8
\(\frac{6+2}{3}=\frac{8}{3}\) (dividing by 3 on both sides)
\(\frac{8}{3}=\frac{8}{3}\) (LHS = RHS)

SOLVING EQUATIONS

To solve an equation means to find a number which when substituted for the variable in the equation, makes its LHS equal to the RHS. This number which satisfies the equation is called the solution or root of the equation.
Example: 2x + 3 = 11
Here, if we consider the value of x = 4,
then 2x + 3 = 11
i. e., LHS = RHS
so, 4 is the root of 2x + 3 = 11.
To find the root (solution) of an equation, i.e., to solve an equation we can follow these methods:
1. Trial and error method
2. Systematic method

Trial and Error Method
In this method, we try different values for the variable to make both sides of an equation equal. We stop this trial as soon as we get a particular value of the variable which makes the LHS equal to the RHS. This particular value is said to be the root of that equation.
Example: Solve 2x – 3 = 5, using trial and error method.
Solution: We try different values of x to find LHS = RHS.
trail-and-error-method
From the above table, we find that LHS = RHS, when x = 4.
∴ Solution is x = 4.

Systematic Method
To solve a linear equation using this method, we add, subtract, multiply, or divide both the sides of the equation by the same number.
Transposing a number (i.e., changing the side of the number) is the same as adding or subtracting the number from both sides of the equation. In doing so, we change the sign of the number.

Example 1: Solve: 2m – 12 = 18.
Solution:
systematic-method
Example 2: A number increased by 15 gives 23. Find the number.
Solution: Let the number be x.
Number increased by 15 = x + 15.
It is 23.
∴ The equation is x + 15 = 23
or, x + 15 – 15 = 23 – 15
(Subtracting 15 from both sides)
or, x = 8
So, the number = 8.

Example 3: Three times a number decreased by 6 gives 12. Find the number.
Solution: Let the number be x.
∴ 3 times of x = 3x
3x decreased by 6 = 3x – 6.
It is 12.
∴ The equation is 3x – 6 = 12
3x – 6 + 6 = 12 + 6
(By adding 6 on both sides)
or, 3x = 18
3x/3 = 18/3
(Dividing by 3 on both sides)
or, x = 6
So, the number = 6